HMMT November

Algebra Difficulty 4.9 AIME Find the answer

The polynomial $x^{3}-3 x^{2}+1$ has three real roots $r_{1}, r_{2}$ , and $r_{3}$ . Compute $\sqrt[3]{3 r_{1}-2}+\sqrt[3]{3 r_{2}-2}+\sqrt[3]{3 r_{3}-2}$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Let $r$ be a root of the given polynomial. Then $r^{3}-3 r^{2}+1=0 \Longrightarrow r^{3}-3 r^{2}+3 r-1=3 r-2 \Longrightarrow r-1=\sqrt[3]{3 r-2}$ Now by Vieta the desired value is $r_{1}+r_{2}+r_{3}-3=3-3=0$ .

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